Optimal. Leaf size=98 \[ \frac {2 (5 B-2 C) \sin (c+d x)}{3 a^2 d}-\frac {(2 B-C) \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac {x (2 B-C)}{a^2}-\frac {(B-C) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.32, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4072, 4020, 3787, 2637, 8} \[ \frac {2 (5 B-2 C) \sin (c+d x)}{3 a^2 d}-\frac {(2 B-C) \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac {x (2 B-C)}{a^2}-\frac {(B-C) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3787
Rule 4020
Rule 4072
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos (c+d x) (B+C \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx\\ &=-\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) (a (4 B-C)-2 a (B-C) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=-\frac {(2 B-C) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \cos (c+d x) \left (2 a^2 (5 B-2 C)-3 a^2 (2 B-C) \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {(2 B-C) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(2 (5 B-2 C)) \int \cos (c+d x) \, dx}{3 a^2}-\frac {(2 B-C) \int 1 \, dx}{a^2}\\ &=-\frac {(2 B-C) x}{a^2}+\frac {2 (5 B-2 C) \sin (c+d x)}{3 a^2 d}-\frac {(2 B-C) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}\\ \end {align*}
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Mathematica [B] time = 0.65, size = 245, normalized size = 2.50 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (-18 d x (2 B-C) \cos \left (c+\frac {d x}{2}\right )-30 B \sin \left (c+\frac {d x}{2}\right )+41 B \sin \left (c+\frac {3 d x}{2}\right )+9 B \sin \left (2 c+\frac {3 d x}{2}\right )+3 B \sin \left (2 c+\frac {5 d x}{2}\right )+3 B \sin \left (3 c+\frac {5 d x}{2}\right )-12 B d x \cos \left (c+\frac {3 d x}{2}\right )-12 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-18 d x (2 B-C) \cos \left (\frac {d x}{2}\right )+66 B \sin \left (\frac {d x}{2}\right )+24 C \sin \left (c+\frac {d x}{2}\right )-20 C \sin \left (c+\frac {3 d x}{2}\right )+6 C d x \cos \left (c+\frac {3 d x}{2}\right )+6 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-36 C \sin \left (\frac {d x}{2}\right )\right )}{12 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 123, normalized size = 1.26 \[ -\frac {3 \, {\left (2 \, B - C\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (2 \, B - C\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (2 \, B - C\right )} d x - {\left (3 \, B \cos \left (d x + c\right )^{2} + {\left (14 \, B - 5 \, C\right )} \cos \left (d x + c\right ) + 10 \, B - 4 \, C\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 121, normalized size = 1.23 \[ -\frac {\frac {6 \, {\left (d x + c\right )} {\left (2 \, B - C\right )}}{a^{2}} - \frac {12 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{2}} + \frac {B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.10, size = 149, normalized size = 1.52 \[ -\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}+\frac {5 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {3 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}+\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 191, normalized size = 1.95 \[ \frac {B {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - C {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.81, size = 109, normalized size = 1.11 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B-C}{a^2}+\frac {3\,B-C}{2\,a^2}\right )}{d}-\frac {x\,\left (2\,B-C\right )}{a^2}+\frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (B-C\right )}{6\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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